
The central torus \(T_\tau :=Z(L_\tau ) \cong \mathbb {G}_m^{l-1}\) , which scales the pieces in the HN graded sheaf, acts trivially on \(Z_\tau \)  but non-trivially on \(X_\tau \) . Thus the \(T_\tau \) -weight picture for \(X_\tau \)  linearised by \(\mathcal {L}_\tau \)  looks like a cone with a vertex at the origin. Consequently semistability equals stability for the action of \(\overline{L_\tau } = L_\tau /\lambda (\mathbb {G}_m)\)  on \(Z_\tau \)  if and only if \(\tau \)  is a coprime HN type (i.e. semistability coincides with stability for all HN subquotients) of length 2. Moreover, for \(l >2\) , the \(\overline{L_\tau }\) -stable locus is empty, which implies that the \(P\) -stable locus in Definition REF  is also empty.

In length \(l >2\) , the unipotent radical \(U_\tau \)  is no longer abelian. Moreover, although the unipotent stabiliser groups often admit sheaf theoretic interpretations, the dimension of these unipotent stabilisers can vary. In particular, for \(l > 2\) , the map \(p_\tau : Y_\tau ^{ss} \rightarrow Z_\tau ^{ss}\)  does not preserve the dimensions of unipotent stabiliser groups in general, thus the blow-up procedure in [1]} is almost intractable as we may blow-up all of \(Z_{\tau }^{ss}\) .

